This module provides a number of homework exercises related to Discrete Random Variables. The module is based on the original module in the textbook collection Collaborative Statistics by Susan Dean and Dr. Barbara Illowsky and has been modified. New problems (#38 to #43) have been added.

1. Complete the PDF and answer the questions.

Xsize 12{X} {}

P(X=x)size 12{P \( X=x \) } {}

X⋅P(X=x)size 12{X cdot P \( X=x \) } {}

0

0.3

1

0.2

2

3

0.4

A

Find the probability that
X=2size 12{X=2} {} .

B

Find the expected value.

A

0.1

B

1.6

Suppose that you are offered the following “deal.” You roll a die. If you roll a 6, you win $10. If you roll a 4 or 5, you win $5. If you roll a 1, 2, or 3, you pay $6.

A

What are you ultimately interested in here (the value of the roll or the money you win)?

B

In words, define the Random Variable
Xsize 12{X} {} .

C

List the values that
Xsize 12{X} {} may take on.

D

Construct a PDF.

E

Over the long run of playing this game, what are your expected average winnings per game?

F

Based on numerical values, should you take the deal? Explain your decision in complete sentences.

A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a software company, has a 10% chance of returning $5,000,000 profit, a 30% chance of returning $1,000,000 profit, and a 60% chance of losing the million dollars. The second company, a hardware company, has a 20% chance of returning $3,000,000 profit, a 40% chance of returning $1,000,000 profit, and a 40% chance of losing the million dollars. The third company, a biotech firm, has a 10% chance of returning $6,000,000 profit, a 70% of no profit or loss, and a 20% chance of losing the million dollars.

A

Construct a PDF for each investment.

B

Find the expected value for each investment.

C

Which is the safest investment? Why do you think so?

D

Which is the riskiest investment? Why do you think so?

E

Which investment has the highest expected return, on average?

B

$200,000;$600,000;$400,000

C

third investment

D

first investment

E

second investment

A theater group holds a fund-raiser. It sells 100 raffle tickets for $5 apiece. Suppose you purchase 4 tickets. The prize is 2 passes to a Broadway show, worth a total of $150.

A

What are you interested in here?

B

In words, define the Random Variable
Xsize 12{X} {} .

C

List the values that
Xsize 12{X} {} may take on.

D

Construct a PDF.

E

If this fund-raiser is repeated often and you always purchase 4 tickets, what would be your expected average winnings per game?

Suppose that 20,000 married adults in the United States were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let
Xsize 12{X} {} = the number of children

Xsize 12{X} {}

P(X=x)size 12{P \( X=x \) } {}

X⋅P(X=x)size 12{X cdot P \( X=x \) } {}

0

0.10

1

0.20

2

0.30

3

4

0.10

5

0.05

6 (or more)

0.05

A

Find the probability that a married adult has 3 children.

B

In words, what does the expected value in this example represent?

C

Find the expected value.

D

Is it more likely that a married adult will have 2 – 3 children or 4 – 6 children? How do you know?

A

0.2

C

2.35

D

2-3 children

Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given below.

Xsize 12{X} {}

P(X=x)size 12{P \( X=x \) } {}

3

0.05

4

0.40

5

0.30

6

0.15

7

0.10

A

In words, define the Random Variable
Xsize 12{X} {} .

B

What does it mean that the values 0, 1, and 2 are not included for
Xsize 12{X} {} on the PDF?

C

On average, how many years do you expect it to take for an individual to earn a B.S.?

Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.

Carole

I'm 13 and I understand it great

AJ

I am 1 year old but I can do it! 1+1=2 proof very hard for me though.

Atone

hi

Adu

Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.

Vedant

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